By Milgram R. (ed.)

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**Extra info for Algebraic and Geometric Topology, Part 2**

**Sample text**

Then it is simple to check that inversion in Σ maps 0 to a. 3 Hyperbolic geodesics Between any two distinct points of B 3 there is an unique path with shortest hyperbolic length. This is a section of a circle orthogonal to the unit sphere S 2 . Lecture 15 57 Proof: Suppose that a and b are two distinct points in the ball B 3 . 1 . For any other value of a, the lemma shows that there is an inversion J in a sphere orthogonal to S 2 with J(a) = 0. Since this inversion is a hyperbolic isometry, we see that the shortest path from a to b must be the image of a radial path under J.

The group G is an infinite dihedral group D∞ . Such a G is the symmetry group of a frieze pattern such as: (c) G = {I, M }. G must contain at least one isometry A with φ(A) = M . This means that A is either a reflection in a line parallel to w or a glide reflection parallel to w. In either case, the other isometries of G that φ maps to M are T k A. Choose co-ordinates so that the origin is on the mirror. In the first case G contains the translations T k , the reflection A, and the glide reflections T k A.

R3+ . 1 Examples in Hyperbolic Geometry The hyperbolic metric is given by: ds = 2 ||dx|| 1 − ||x||2 on B 3 and ds = 1 ||dx|| x3 on R3+ . The hyperbolic geodesics are the arcs of circles orthogonal to the boundary ∂H3 . The plane {x : x3 = 0} meets the ball B 3 in the unit disc and the hyperbolic metric on B 3 restricts to the hyperbolic plane metric on this disc. A similar result holds for the intersection of any other sphere orthogonal to ∂H3 with H3 . For suppose that Σ is the intersection with B 3 of a sphere orthogonal to ∂B 3 .